1. ## differentiate

differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks

2. ## Re: differentiate

Originally Posted by Plato13
differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks
Note that $\displaystyle | w + t |^2 = (w + t)^2$ is always positive. Thus
$\displaystyle \frac{w + t}{ |w + t| ^2 } = \frac{w + t}{(w + t)^2}$

What can you do with this?

-Dan

3. ## Re: differentiate

W is a complex number so this is not true.

4. ## Re: differentiate

Originally Posted by Plato13
differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks
Recall that $\displaystyle |z|^2=z\cdot\overline{z}$ and because $\displaystyle t$ is real then $\displaystyle \overline{t}=t.$

Thus $\displaystyle |w+t|^2=(w+t)\overline{(w+t)}=|w|^2+2\text{Re}(w)t +t^2~$. So$\displaystyle \frac{d(|w+t|^2)}{dt}=2\text{Re}(w)+2t$

5. ## Re: differentiate

Originally Posted by Plato
Recall that $\displaystyle |z|^2=z\cdot\overline{z}$ and because $\displaystyle t$ is real then $\displaystyle \overline{t}=t.$

Thus $\displaystyle |w+t|^2=(w+t)\overline{(w+t)}=|w|^2+\text{Re}(w)t+ t^2~$. So$\displaystyle \frac{d(|w+t|^2)}{dt}=\text{Re}(w)+2t$
Plato, shouldn't that be 2Re(w)t?

6. ## Re: differentiate

Originally Posted by Plato13
Plato, shouldn't that be 2Re(w)t?
Right! I will edit.

7. ## Re: differentiate

Thanks for the help